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## Homework Statement

Find the transformation matrix R that describes a rotation by 120 about an axis from the origin through the point (1,1,1). The rotation is clockwise as you look down the axis toward the origin.

## Homework Equations

Rotations about the z-axis are given by

[tex]R_{z}(\alpha) = \left( \begin{array}{ccc}

cos(\alpha) & sin(\alpha) & 0 \\

-sin(\alpha) & cos(\alpha) & 0 \\

0 & 0 & 1

\end{array} \right)[/tex]

whereas rotations about the x-axis are given by

[tex]R_{x}(x) = \left( \begin{array}{ccc}

1 & 0 & 0 \\

0 & cos(x) & sin(x) \\

0 & -sin(x) & cos(x)

\end{array} \right)[/tex].

## The Attempt at a Solution

My strategy in solving this problem was to rotate the coordinate system in such a way as to align the z-axis along the axis extending from the origin to (1,1,1). Once this was done, I was to rotate the system as a regular rotation in a two-dimensional x-y system.

The first rotation should be such that the x-axis is aligned perpendicular to the x-y projection of [tex]\hat{x} + \hat{y} + \hat{z}[/tex], or perpendicular to [tex]\hat{x} + \hat{y}[/tex]. This was done with a rotation about the z-axis, more specifically [tex]R_{z}(\frac{3 \pi}{4})[/tex].

I intended the second rotation to be about the x-axis to orient the z-axis as desired. Working now with primed coordinates after the previous rotation the desired axis lied in the y'-z plane. The coordinates of the original vector <1, 1, 1> in the primed system was [tex]\sqrt{2} \hat{y} + \hat{z}[/tex]. Therefore, I wanted to rotate the x-axis clockwise by [tex]Cos^{-1}(\frac{1}{\sqrt{3}})[/tex] degress. However, the way my matrices in section 2 were set up should have all rotations going counterclockwise, so I wanted my rotation matrix to be [tex]R_{x}(2 \pi - Cos^{-1}(\frac{1}{\sqrt{3}}))[/tex].

Now that the z-axis was properly aligned, I could rotate about it, so my final rotation matrix should be [tex]R_{z}(\frac{2 \pi}{3})[/tex].

If my logic is correct then the final rotation should be

[tex]R = R_{z}(\frac{2 \pi}{3}) * R_{x}(2 \pi - Cos^{-1}(\frac{1}{\sqrt{3}})) * R_{z}(\frac{3 \pi}{4})[/tex].

That said, I know my answer should be

[tex]R = \left( \begin{array}{ccc}

0 & 0 & 1 \\

1 & 0 & 0 \\

0 & 1 & 0

\end{array} \right)[/tex]

however this is not what I am getting. I am getting something very messy. Where have I gone wrong?